读书笔记：算法导论第2章 第1节 Insertion sort

1. We use loop invariants to help us understand why an algorithm is correct. We must show three things about a loop invariant:

• Initialization: It is true prior to the first iteration of the loop.
  
• Maintenance: If it is true before an iteration of the loop, it remains true before the next iteration.
• Termination: When the loop terminates, the invariant gives us a useful property that helps show that the algorithm is correct.

2. Exercise2.1-1:

Question: Insertion Sort  a[] = {31, 41, 59, 26, 41, 58} by increasing order.

Source code:

#include <stdio.h>void insertionSort(int *a, int len){    int i, j, key;    for(i = 1; i < len; i++){        key = a[i];        j = i - 1;        while(j >= 0 && a[j] > key){            a[j + 1] = a[j];            j--;        }        a[j + 1] = key;    }}int main(int argc, char *argv[]){    int a[] = {31, 41, 59, 26, 41, 58};    int len = sizeof(a)/ sizeof(a[0]);    int i;    printf("Before insertion sort:\n");    printf("a[] = {");    for(i = 0; i < len - 1; i++)        printf("%d, ", a[i]);    printf("%d}\n", a[len - 1]);    insertionSort(a, len);    printf("After insertion sort:\n");    printf("a[] = {");    for(i = 0; i < len - 1; i++)        printf("%d, ", a[i]);    printf("%d}\n", a[len - 1]);    return 0;}

3.Exercise2.1-2:

Question:Rewrite the INSERTION -SORT procedure to sort into nonincreasing instead of decreasing order.

Source code:

void insertionSort(int *a, int len){    int i, j, key;    for(i = 1; i < len; i++){        key = a[i];        j = i - 1;        while(j >= 0 && a[j] < key){            a[j + 1] = a[j];            j--;        }        a[j + 1] = key;    }}

4.Exercise 2.1-3

Question:

Consider the searching problem:
Input: A sequence of n numbers A={a1,a2, ...,an} and a value v.
Output: An index i such that v = a[i] or the special value NIL if v does not appear in A.Write pseudocode for linear search, which scans through the sequence,
looking for v. Using a loop invariant, prove that your algorithm is correct. Make sure that your loop invariant fulfills the three necessary properties.

pseudo code:

LINEAR-SEARCH(A):
1   for i = 0 to A.length - 1
2          if v == A[i]
3                print  i
4   if i==A.length
5          v == NIL

5. Exercise 2.1-4
Question: Consider the problem of adding two n-bit binary integers, stored in two n-element arrays A and B.
The sum of the two integers should be stored in binary form in an (n + 1)-element array C .

Source code:

#include <stdio.h>void binary_array_addition(int *a, int *b, int *c, int len){    int temp, carry, i;    carry = 0;    for(i = len - 1; i >= 0; i--){        temp = a[i] + b[i] + carry;        if(temp == 3){            carry = 1;            c[i + 1] = 1;        }        else if(temp == 2)            carry = 1;        else if(temp == 1){            carry = 0;            c[i + 1] = 1;        }    }    if(carry == 1)        c[i + 1] = 1;}int main(int argc, char *argv[]){    int i;    int a[] = {1, 0, 1, 1, 0};    int b[] = {1, 1, 0, 1, 1};    int len = sizeof a / sizeof a[0];    int c[sizeof a / sizeof a[0] + 1] = {0};    binary_array_addition(a, b, c, len);    printf("a[] + b[] = ");    for(i = 0; i < len; i++)        printf("%d", a[i]);    printf(" + ");    for(i = 0; i < len; i++)        printf("%d", b[i]);    printf(" = ");    for(i = 0; i < len + 1; i++)        printf("%d", c[i]);    putchar('\n');    return 0;}